All Questions
663 questions with no upvoted or accepted answers
2
votes
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532
views
The Sato-Tate conjecture (Frobenius eigenvalues)
Another question crossed my mind.
In the statement of the Sato-Tate conjecture, one usually assumes that
the elliptic curve has no CM. But, I read the Morita's paper for the BSD in the CM
case and ...
- 342
2
votes
0
answers
840
views
Elliptic curves with potential good reduction over a prescribed extension
Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ ...
- 998
2
votes
0
answers
215
views
Generalized "elliptic integrals"
I am interested in evaluating the following type of integrals. Here we a polynomial $q(x)$ of degree $d \geq 2$ with no non-negative roots. Then is there a name for integrals of the shape
$$\int_0^\...
- 27k
2
votes
0
answers
245
views
Help for reference of moduli stack of fake elliptic curves
I see everywhere the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.
...
- 21
2
votes
0
answers
424
views
Number of CM lifting of an ordinary elliptic curve
Before asking my questions I will start with an example: There are two CM elliptic curves over $\mathbb{Q}$ with CM field $\mathbb{Q}(\sqrt{-7})$, whose $j$-invariants are $-3^3.5^3$ and $3^3. 5^3. 17^...
- 21
2
votes
0
answers
182
views
An elliptic curve trivial over any extension unramified outside 7 and infinity?
Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
- 11.3k
2
votes
0
answers
300
views
padic BSD vs. BSD for algorithm to compute rank
Just to be specific, I deal only with the elliptic curves $E$ over $\mathbb{Q}$, and most of the explaination here are obtained from the paper: Algorithms for the Arithmetic of Elliptic Curves using ...
- 173
2
votes
0
answers
326
views
Proportion of rational elliptic curves of a given rank
This morning appeared on Arxiv the following article by Manjul Bhargava et al: http://arxiv.org/pdf/1407.1826.pdf, in which the authors give a lower bound for th proportion of rational elliptic curves ...
- 6,982
2
votes
0
answers
119
views
Elliptic surfaces with different Kodaira symbols
Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre?
Can ...
- 133
2
votes
0
answers
430
views
Average rank of elliptic curves over function fields
De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...
- 809
2
votes
0
answers
311
views
Finite Heisenberg groups action on cohomology of line bundles
Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the ...
- 1,545
2
votes
0
answers
7k
views
Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve
$\DeclareMathOperator\Aut{Aut}\newcommand{\alg}{\mathrm{alg}}\newcommand{\an}{\mathrm{an}}$Edited after Noam Elkies' comment: From what I understand (very little actually), there exist elliptic curves ...
- 6,982
2
votes
0
answers
244
views
Descent theory of line bundles on abelian varieties under isogenies (in char p>0)
I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
- 614
2
votes
0
answers
256
views
Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$
Hi, overflowers.
I have a question concerning the torsion of elliptic curves over number fields.
Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can ...
- 306
2
votes
0
answers
255
views
Lang isogeny for group stacks
Let $G$ be a commutative algebraic group stack over $\mathbb{F}_q$ (I don't really care about the precise definition: I'm secretly thinking about the Picard stack of a projective curve). To what ...
- 3,605
2
votes
0
answers
177
views
A nice rigid analytic model for local systems over an elliptic curve?
For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...
- 7,972
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
2
votes
0
answers
190
views
Curve C of genus 2 whose equation satisfies equation in Igusa invariants, but where Jac(C) does not split
The background question is: Let $C$ be a curve of genus $2$ over a field $k$. When is there a degree $2$ morphism from $C$ to an elliptic curve (and therefore an isogeny from the Jacobian $\mbox{Jac}(...
- 121
2
votes
0
answers
632
views
Kernel of an \'etale isogeny of prime degree $\ell$ between elliptic curves
I recently try to read Vatsal's paper ``Multiplicative subgroups of $J_0(N)$ and applications to elliptic curves.'' He seemed to use the following fact freely:
Let $E$ be a semistable elliptic ...
- 21
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
user19475
2
votes
0
answers
234
views
How does one get that for elliptic curves $A$, $B$ over a number field, $a_\mathfrak{l}=b_\mathfrak{l}$ implies that the curves are isogenous?
Let $K$ be a number field, and let $A$ and $B$ be two elliptic curves over $K$. For a nonzero prime ideal $\mathfrak{l}$ of $K$, outside a finite set of primes, let $a_\mathfrak{l}$ and $b_\mathfrak{l}...
- 2,406
2
votes
0
answers
310
views
Moduli space of points of fixed order N on elliptic curves
Let us consider the moduli space $Y_1(N)$, parametrizing an elliptic curve, together with a choice of a point of N-torsion on it. Over the complex numbers, it is easy to see that this moduli space is ...
- 63
2
votes
0
answers
415
views
bounds on regulator of elliptic curve,
Let E be an elliptic curve over Q with positive rank and trivial torsion structure. Is there any sort of upper bound (conjectural or unconditional) on the regulator of E in terms of the conductor of E?...
- 818
2
votes
0
answers
321
views
Dimension of fibres of moment maps in characteristic $p$
Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that ...
- 3,545
2
votes
0
answers
381
views
elliptic curves with CM and hecke L-series
I know that if you have an elliptic curve E with complex multiplication, then the Hasse Weil L-series attached to it can be expressed in terms of Hecke L-series. Is there anything that can be said ...
- 21
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,006
1
vote
0
answers
137
views
Two elliptic curves with the same j-invariants
This is an interesting observation of mine when exploring moduli of elliptic curves.
Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
- 81
1
vote
0
answers
157
views
Degeneracy of the Cassels-Tate pairing $\operatorname{Ш}(E/K)[n]\times \operatorname{Ш}(E/K)[n]\to \Bbb{Q}/\Bbb{Z}$
$\DeclareMathOperator{\Sha}{Ш}$ Let $E/K$ be an elliptic curve over a number field $K$. Let $\Sha (E/K)$ be the Tate-Shafarevich group, and let $n\ge 2$ be an integer. According to Theorem 15 in the ...
- 1,541
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
1
vote
0
answers
69
views
Is it in theory possible to perform general Miller’s algorithm inversion as used with the optimal ate pairing with large trace in subexponential time?
Let’s I have the following :
2 curves $G_1$ defined on $F_p$ and $G_2$ being the $G_1$ curve’s twist defined on $F_p^2$ both having the same prime order ; a large trace ; and $F_p^{12}$ as their ...
- 87
1
vote
0
answers
85
views
Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
1
vote
0
answers
153
views
Tate curve and components of special fibre
Let $K$ be a complete local field of characteristic $0$ with ring of integers $R=\mathcal{O}_{K}$ and residue field $k=R/\mathfrak{m}_R$ of characteristic $p>0$. Let $E/K$ be an elliptic curve of ...
- 6,006
1
vote
0
answers
115
views
Compactifications of product of universal elliptic curves
Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving ...
- 12.1k
1
vote
0
answers
157
views
Does this subset of elliptic curves over $\mathbb{Q}$ have positive proportion?
Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times ...
- 61
1
vote
0
answers
100
views
Period lattice of cubic twists of CM modular forms
Let $K=\mathbb{Q}(\sqrt{-3})$ be a CM field. Let $E_1:y^2=x^3+1/4$ and $E_p:y^2=x^3+p^2/4$ where $p\equiv 1\mod 3$ is a prime. Let $f_1$ and $f_p$ be the modular forms of $E_1$ and $E_p$. They are ...
- 390
1
vote
0
answers
76
views
Global minimal discriminants of elliptic curves and Galois representations
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if ...
- 1,428
1
vote
0
answers
143
views
Describing the hyperbolic structure of punctured torus in terms of the period lattice
Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...
- 7,547
1
vote
0
answers
90
views
Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
- 121
1
vote
0
answers
169
views
Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
- 6,006
1
vote
0
answers
103
views
Criterion for an etale cover $E[\ell]\to \mathbb{G}_m$ to be tamely ramified in $0, \infty$
Suppose $E\to \mathbb{G}_m/k$ is an elliptic curve with $k$ field of characteristic $p>0$ and $E[m]$ it $m$-torsion group with $(m,p)=1$.
Consider the induced finite etale cover $E[\ell]\to \mathbb{...
- 6,006
1
vote
0
answers
102
views
Tate-Shafarevich group and its twist such that $\text{Sha}(E_D/\Bbb{Q})=0$ or some constant
Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$.
Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist.
It is widely known that for all $E/\Bbb{Q}$: elliptic ...
- 1,541
1
vote
0
answers
133
views
Select random point on elliptic curve
If I have an elliptic curve $E$ over some finite field $F_p$ what is a step by step algorithm to pick a random point that lays on this curve? There is definitely a naive approach to brute force all ...
- 11
1
vote
0
answers
90
views
Isogenous elliptic curves in characteristic zero and in characteristic $p$
Assume two elliptic curves (with CM), $E_{1}$ and $E_{2}$, are isogenous over a field $K$ of characteristic zero. Are the following two statements true?
(a) Their $V_{p}$ modules are $G_{K}$-...
- 71
1
vote
0
answers
120
views
Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$
Let $p$ be a prime number and $n$ be positive integer.
Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.
LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$.
This is the biggest size of $Sha(...
- 1,541
1
vote
0
answers
59
views
Density of integer representations of a a two variable polynomial
I was testing some data on the positive integer representations of $f(x, y) = x^3 - y^2$ such that $(x, y) \in [1, M] \times [1, M]$. I tested it for $M = 1000, 10000$. It turns out that the density ...
- 577
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
1
vote
0
answers
162
views
Exercise in Cassels's book
I am trying to prove the following theorem:
Theorem. Let $d=q_1q_2$ where $q_1> 0$, $q_2>0$ are rational primes, with $q_1\equiv 2\mod 9$ and $q_2 \equiv 5 \mod 9$. Then the only rational point ...
- 11
1
vote
0
answers
320
views
Tate-Shafarevich group of Elkies curve
The Elkies curve
$$
E:y^2+xy+y=x^3−x^2−20067762415575526585033208209338542750930230312178956502x+34481611795030556467032985690390720374855944359319180361266008296291939448732243429
$$
conductor of ...
- 245
1
vote
0
answers
91
views
Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?
Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...
- 1,833